A principalization algorithm for locally monomial ideal sheaves on 3-folds with an application to toroidalization

Suppose that I is an ideal sheaf on a nonsingular variety X. A principalization of I is a proper birational morphism λ:X˜→X such that X˜ is nonsingular and IOX˜ is locally principal. We provide a fast and simple algorithm to construct a principalization of a locally monomial ideal sheaf on a 3-fold. As an application we prove the existence of toroidalization of locally toroidal morphisms of 3-folds over an algebraically closed field of characteristic zero.